Bivariate Bernstein Chlodovsky Operators Preserving Exponential Functions and Their Convergence Properties

Abstract

This paper is devoted to an extension of the bivariate generalized Bernstein-Chlodovsky operators preserving the exponential function exp ( 2 , 2 ) where exp ( α , β ) = e − α x − β y , α , β ∈ R 0 + and x , y ≥ 0 . For these operators, we first examine the weighted approximation properties for continuous functions in the weighted space, and in the latter case, we also obtain the convergence rate for these operators using a weighted modulus of continuity. Then, we investigate the order of approximation regarding local approximation results via Peetre’s K-functional. We introduce the GBS (Generalized Boolean Sum) operators of generalized Bernstein-Chlodovsky operators, and we estimate the degree of approximation in terms of the Lipschitz class of Bögel continuous functions and the mixed modulus of smoothness. Finally, we provide some numerical and graphical examples with different values to demonstrate the rate of convergence of the constructed operators.